This invention was conceived as way of adding field lines to the Mandelbrot and Julia set images. The underlying invention also serves as a general method of categorizing escape regions of complex and hypercomplex chaotic functions and fractals.
1. Field of Invention
This invention generally relates to the chaotic functions and Julia set fractals which are generated from numerically iterating simple complex arithmetic functions.
2. Prior Art
Prior Art for Interval Processing
Overview
Prior art does not provide complete system for optimal interval with stable numerical algorithms.
Accuracy and Stability of Numerical Algorithms, 1996
    Accuracy and Stability of Numerical Algorithms, by Nicholas J. Higham, Copyright 1996, Society for Industrial and Applied Mathematics (SIAM), Philadelphia. ISBN 0-89871-355-2
The book describes numerical error analysis. Its analysis of rounding error for complex numbers was important to understanding a similar analysis for quaternions.
C++ Toolbox for Verified Computing, 1995
    C++ Toolbox for Verified Computing, by Ulrich Kulisch, Rolf Hammer, Matthias Hocks, Dietmar Ratz, Copyright 1995, Springer-Verlag, Berlin, Heidelberg. ISBN 3-540-59110-9 Springer-Verlag Berlin, Heidelberg, and New York.
The book describes the use of interval processing in mathematical proofs for parametric spaces. The scope is general, but it does not provide a description of optimal complex intervals.
Complex Interval Arithmetic and Its Applications, 1995
    Complex Interval Arithmetic and Its Applications, by Miodrag S. Pekovic and Ljiljana D. Petkovic, Copyright 1998, WILEY-VCH, Berlin; Weinheim; New York; Chichester; Brisbane; Singapore; Toronto; ISBN 3-527-40134-2 Springer-Verlag Berlin, Heidelberg, and New York.Overview
This book emphasizes the use of optimal complex arithmetic on discs and many applications, and discusses the not so optimal. Techniques for optimal arithmetic and integer powers for complex numbers are described. This book discusses additional optimal disc forms which are not improved upon in this patent application, such as logarithmic, exponential, and some other transcendental functions.
Optimal Multiplication Problems
Overview
The book provides several references to a N. Krier, a German who describes in doctoral paper, apparently in German, and a method of performing optimal complex disc multiplications.
The Krier formula given in this book (I don't know about the original Krier work), does not appear to produce the correct answers—see page 23. See also page 82 for additional discussion of the Krier results that are useful. (See my own invention which agrees with results for page 82). The authors note several shortcomings in the use of the formulas, and thus discourage use of this powerful tool.
Optimal Multiplication Computation Problem
First, at the bottom of page 23, it is noted that “the presented minimal arithmetic requires a great deal of computation, including the calculation of the zero of third degree polynomial.” No solution to this problem is provided.
Inclusion Isotonicity Property Problem
The authors immediately continue with a second problem at the very bottom of page 23, “Besides, a very important inclusion isotonicity property is not generally valid for multiplication (and, consequently, for division (1.15))”. No solution to this problem is provided, but an example of the problem is indicated on page 24.
Real Powers
The book develops algorithms for finding optimal disc enclosures for positive integer powers. Optimal discs for real powers, 1/k, where k is an integer are presented. The authors fail to create a comprehensive real power interval operator for all real powers. Also, on page 82, notes that squaring a disc interval is equivalent to Krier's multiplication when squaring. The author does not work out equations to provide an explicit formula for squaring.
Quaternions and Rotation Sequences, 1999
    Quaternions and Rotation Sequences, by Jack B. Kuipers, Copyright 1999, by Princeton University Press, Princeton, N.J., ISBN 0-691-05872-5.
This book provides background on the arithmetic of quaternions, but no interval analysis or details on numeric processing. Quaternions are important example of hypercomplex numbers of the 4th dimension; however, but there is no discussion of hypercomplex numbers of other dimensions.
Prior Art for this Application
Books on Chaotic Functions
The Beauty of Fractals, 1986
    The Beauty of Fractals, by Heinz-Otto Peitgen and Peter H. Richter, Copyright 1986 by Springer-Verlag ISBN 3-540-15851-0 Springer-Verlag Berlin, Heidelberg, New York, and Tokyo, ISBN 0-387-15851-0 Springer-Verlag Berlin, Heidelberg, New York, and Tokyo.
This early book describes the fundamentals of discrete dynamic systems, with emphasis on Mandelbrot-like iterations. This book emphasizes visualization as a method of investigating and exploring topological properties.
Pixel Categorization Problem
For processing an entire pixel, is that although the visual field has an infinite number of parameter points per pixel, only a finite number of parameter points are actually processed. The book fails to provide an absolute method of proving an entire pixel to be part of a topological category, such as escaping the Mandelbrot. A particularly sharp “spike” of the Mandelbrot boundary could be missed by the computer graphic pointwise generation.
Potential
For many functions similar to the Mandelbrot exterior, it is possible to assign a “potential” to represent the speed at which the absolute value of pointwise iteration escapes to infinity.
Potential Commonality Problem
For the purposes of visualization, one can color code a point based on the number of iterations to achieve a particular absolute cutoff distance; this is apparently done in the book to great effect. (On page 163 a more formal method is presented, based on absolute value divided by 2 to the power “n”, where n is the number of iterations to achieve it. This method forms the basis for many visualizations in later literature, but the author notes the sensitivity of small changes in the algorithm to affect the aesthetics). A general approach to iterated functions is not broached.
Field Line Generation
On page 68, for the referenced formula called (5.11), the text says that the formula was used to calculate FIG. 16 of the same book. FIG. 16 of the book appears to be an artistic line drawing of potential and field lines. Its unclear if the formula was used for field lines or just potential lines. In any case, the formula cannot be directly applied as a solution to field lines.
Field Line Problem
Although the book describes the computer visualization of potential fields of discrete dynamic systems, its visualization of field lines is largely left to artistic renderings. See FIG. 16. on page 16. Field lines which are shown in the outermost area of the Mandelbrot are computer generated, but use a scheme of “Binary Expansion”, which is, apparently, only an approximation, which works when not too close to topological boundaries. See page 66. Note dramatic image on page 74.
With half of the complex value missing, namely the field line value, there is no way to use all the advanced techniques of computer graphics for texture mapping and surface modeling. Therefore, although this book is famous for its wondrous beauty of chaos, there is an inherent limitation to all renderings.
The Science of Fractal Images, 1988
    The Science of Fractal Images, by Heinz-Otto Peitgen and Dietmar Saupe (Editors), Copyright 1988 by Springer-Verlag ISBN 3-540-96608-0 Springer-Verlag Berlin, Heidelberg, and New York, and ISBN 0-387-96608-0 Springer-Verlag Berlin, Heidelberg, and New York.Topological Categorization Proofs: Koebe ¼ Theorem
The book succeeds in finding a direct method of proving entire areas to be external, or escaped, for the Mandelbrot, with analysis dependent upon the Koebe ¼ Theorem. The method is discussed in detail in Appendix D on page 287. This landmark contribution allows areas of visualization to be quickly categorized as external. Elaborate mathematical proofs are described to assure the reader that the method is correct. This method uses a calculation of the magnitude of the derivative to find a bounds between the minimum and maximum distance between an external point and the boundary of a Mandelbrot-like iteration. Only the minimum value is utilized, as it provides a disc around the point where all points are guaranteed to be external (e.g. escaped). Thus, large areas of points are all guaranteed as escaped, by a single iteration. On page 292, images show the dramatic reduction in pointwise iterations in order to categorize areas exterior to the Mandelbrot.
Koebe Categorization solves Pixel Categorization Aliasing Problem.
This approach, when applicable, can be used to solve the Pixel Categorization Aliasing Problem of the earlier book, The Beauty of Fractals. When used with standard interval analysis, guaranteed categorizations are available.
Koebe Distance Applicability Problem
Although the Mandelbrot and some Julia sets are suitable for distance estimates. Other functions may not have such simple methods. Thus the method for bounding topological distances may be difficult to apply in specific circumstances.
Field Line Generation:
On page 197, in a figure called 4.17, a drawing is given for field lines. I could not find a reference to its generation. However, this zoomed-out view of the Mandelbrot field lines is typical; field lines deep inside the tangles of the Mandelbrot are not shown. This book also shows the use of binary decomposition to construct field lines (but as always, only for zoomed-out images).
Chaos and Fractals, The Mathematics Behind the Computer Graphics 1989
    Chaos and Fractals, The Mathematics Behind the Computer Graphics, by Robert L. Devaney and Linda Keen, Copyright 1989 by the American Mathematical Society, ISBN 0-8218-0137-6.
This book is mentioned because of its artistic rendering of external field lines.
Computers, Pattern, Chaos, and Beauty, 1990
    Computers, Pattern, Chaos, and Beauty, by Clifford A. Pickover, Copyright 1990 and 2001 by Dover Publications, Inc., Mineola, New York, ISBN 0-486-41709-3,
The book describes is a more general book on the subject, but offers some interesting, and very simple, modifications to the escape algorithms for discrete dynamic systems.
Topological Categorization Proofs
Controlled Step Escape Modification
Earlier works on discrete dynamic systems were beholding to early termination of infinite iterations. These earlier models also required some way of ascertaining an escape distance which would guarantee correct categorization without an infinite number of iterations. The book's modifications allow the escape process to terminate at an exact iteration stage and still produce visually exciting results that take on the appearance of “organisms”. See chapter 8. Most previous “escape” algorithms rely on testing the absolute value of a complex iterated point. Here, the test does something different, although still based loosely on real and imaginary parts of the absolute value function. This method is very effective and is described section 8.2, entitled “A Computer Program ‘Bug’”. This approach produces new results and offers a lot more bang for the “buck” because “escape” testing is greatly simplified.
Chaos and Fractals: new frontiers of science, 1992
    Chaos and Fractals, by Heinz-Otto Peitgen and Dietmar Saupe, and Hartman Jürgens, Copyright 1992 by Springer-Verlag, New York Inc., New York, ISBN 3-540-97903-4 Springer-Verlag Berlin, Heidelberg, and New York, and ISBN 0-387-97903-4 Springer-Verlag Berlin, Heidelberg, and New York.
This huge book, 984 pages, takes a broad look at chaos theory.
Field Lines
Chapter 13 provides extensive discussion of binary expansion, as was described in the book The Beauty of Fractals, above. Although extensive, it fails to address the problem of the method's approximation effect which breaks down as one delves deeper at higher magnifications near the boundary of the Mandelbrot.
Hypercomplex Systems
Page 837 describes the use of higher dimensional processing for Julia sets, but is easily applicable to the 191 Mandelbrot. Color plates 15 and 16 show color renderings of a Julia set in three dimensions.
Hypercomplex Iterations, Distance Estimation and Higher Dimensional Fractals, 2002
    Hypercomplex Iterations, Distance Estimation and Higher Dimensional Fractals, by Yumei Dang, Louis H. Kauffman, and Dan Sandin, Copyright 2002 by World Scientific Publishing Co. Pte. Ltd., Singapore, ISBN 981-02-3296-9
The book describes, in great detail, the application of a distance to boundaries for Mandelbrot and Julia sets. The method here is an extension of that found in the book The Science of Fractal Images, above. Higher dimensional iterations are emphasized.
Hypercomplex Systems and Topological Proofs
The book presents additional theory and some practical aspects of using distance estimation in higher dimensions. Although the additional theory is useful, but application of these techniques are limited to a small variety of Julia sets and Mandelbrot-like iterations.
Books on Complex Analysis
Visual Complex Analysis, 1997
    Visual Complex Analysis, by Tristin Needham, Copyright 1997, by Clarendon Press, Oxford, ISBN 981-02-3296-9.
This book was used to research winding numbers, which are simply a way of understanding the number of times a complex path wraps around complex origin.